Mesh with Triangulate Modifier mesh causes new faces and vertices to form with the Subsurface Modifier and Bevel Modifier
If you don't want to triangulate the faces manually, you can use the triangulate modifier. Then, add a bevel modifier, and a subdivision surface modifier:



This gives the following result:



Be sure to triangulate before subdividing!

I'm noticing that some of the circles are still "blocky". To fix this, try to go back, before you beveled it, and use a bevel modifier (not to be confused with the one above), setting a limit on the angle. Example:

No limit:



With limit (note that the less pointy edges were not affected, and will be smoothed out by a subdivision surface modifier):